540 PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
Thus vre shall have, 
(l(^ 
r=A— «(E— cos (NT+s — D) — ^ cos 2(N^-l-2 — n)+&c. 
-i2.Fcosi-(NT+2-N'^-£') 
2.G cos{A’(N^-i-£ — — a')-|-N^-l-s— ri} 
+^2.IIcos{(6--l)(NT-l-£-N7-a')+N^+s-n'}. 
And putting F', G', IT for the coefficients of sin^^', sin and sin in 
the development of d, we have 
5 ^ 
^ = a+N^+2(E + ^) sin (N^+s — 11)+-^ sin 2(NT+e — n)+&c. 
+ e'g sin(N^+£ — 11')+;^ 2 . F'sin .v(N^+£~N'^ — a') 
2 . G'sin {^(N^+a — N'^— a')+N^-f-a — 11} 
2 . H'sin — l)(N^-l-a — N'^— a')+N^-}-a — H'}. 
17. I proceed next to draw some conclusions from these values of the radius- 
vector and longitude. 
(1) The quantity A is the non-periodic part of the radius-vector, and being equal 
X d a 
toa — is a function of given quantities and arbitrary constants. A is, 
therefore, invariable. It may also be remarked, that as the value of r may be put to 
the same approximation under the form 
" ( ' -i - v+ ^ 
1 dh. 
the quantity ^ is approximately the mean distance. Thus, so far as this 
approximation shows, the mean distance is invariable. 
(2) The mean motion is necessarily the factor of the non-periodic term Nf in the 
development of 6. Hence 
n/r • -IWT .2 «?Ao 
Mean motion =N=w4 — - 1 — 
' na da 
For the reason just adduced, the mean motion is thus proved to be invariable. 
As the two quantities A and N are functions of a and e, they are by consequence 
functions of the arbitrary constants h and C. Hence, if the values of the non- 
periodic part of the radius-vector and the mean motion be deduced from observa- 
tion, the constants a and e, or h and C, become known. 
(3) The quantity a, being simply an arbitrary constant, is invariable. Analogous 
considerations apply to the mean distance, mean motion, and the epoch (a') of the 
orbit of m! as disturbed by m. 
